Results Math 72 (2017), 181–192
2016 Springer International Publishing
published online December 23, 2016
Results in Mathematics
Aﬃne Hypersurfaces with Parallel Shape
Abstract. We prove the following: a relative hypersurface with parallel
shape operator is either a relative hypersphere, or it is aﬃnely equivalent
to an example constructed by Th. Binder. Furthermore, based on Binder’s
example, we give another simple and more explicit example; this way we
improve the classiﬁcation and show that it is completely determined by
; t), the latter being solutions of certain Monge–
Amp`ere equations. Our example geometrically is constructed from a plane
curve and a family of relative hyperspheres. In case of an aﬃne sphere
with Blaschke geometry we show that our classiﬁcation can be consid-
ered as a construction coming from a plane curve together with a family
of improper aﬃne hyperspheres. Especially in R
, this construction is
determined by only three functions of a single variable.
Mathematics Subject Classiﬁcation. 53A15.
Keywords. Blaschke hypersurface, improper aﬃne hypersphere,
Monge–Amp`ere equation, parallel shape operator.
Jelonek  proved that the fact that the shape operator of a relative hypersur-
face is parallel to the aﬃne induced connection ∇ is equivalent to the fact that
it is parallel to the Levi-Civita connection. In dimension n = 2 such Blaschke
hypersurfaces have been studied by several authors from diﬀerent point of view
Supported by Fundamental Research Funds for the Central Universities (Grant
No. 20720150009) and Education and Scientiﬁc Research Program for Middle-aged and
Young Teachers of Fujian Province (Grant No. JA15007).